All players dropped from a team's roster are immediately placed on waivers. During this waiver period—two days in Public Leagues and anywhere from zero to seven days in Custom Leagues—anyone in the league can place a claim on the player.
So, if a player in a Public League is dropped on a Wednesday afternoon, all managers can place a claim on him until Friday at 11:59 p.m. PT. The player is then assigned to the new team roughly 4-8 hours after the claim deadline.
To find the length of the waiver period in your league, check the League Settings page, which can be found via a link on your League Overview page.
If more than one manager claims the same player, the team that's highest on the Waiver Priority List is given the rights to the player.
The priority list is initially set based on the reverse of the draft order. However, each time a team is awarded a player via waivers, that team is pushed to the back of the list.
If you're viewing the waiver priority list after a player has been processed, the team with the lowest priority (12 in a 12-team league, 10 in a 10-team league, etc.) is the team that was most recently awarded a player.
Your current position on the waiver priority list is displayed in a column along with your league standings.
If nobody claims a player during the waiver period, he becomes a free agent. In a Custom League with no waiver period, all dropped players automatically become free agents.
If you place a player on waivers, you cannot reclaim that player until after the waiver period has expired.
If you place a claim on more than one waived player at a time, each pending transaction appears on your Team Page along with its current position in your waiver order. All new claims get appended to the end of the list, but you can alter this priority by viewing each pending claim and changing its priority in the box provided.